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Properties

Label 855.2.l.a
Level $855$
Weight $2$
Character orbit 855.l
Analytic conductor $6.827$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [855,2,Mod(391,855)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(855, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("855.391");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.l (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 4 q^{2} - 2 q^{3} + 80 q^{4} - 40 q^{5} - 6 q^{6} + q^{7} + 24 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 4 q^{2} - 2 q^{3} + 80 q^{4} - 40 q^{5} - 6 q^{6} + q^{7} + 24 q^{8} - 10 q^{9} - 2 q^{10} - q^{11} - 6 q^{12} + 14 q^{13} + q^{15} + 72 q^{16} + 6 q^{17} + 11 q^{18} + q^{19} - 40 q^{20} + 4 q^{21} - 15 q^{22} + 34 q^{24} - 40 q^{25} - 8 q^{26} - 2 q^{27} - 5 q^{28} - 3 q^{29} + 9 q^{30} - 16 q^{31} + 52 q^{32} - 35 q^{33} + 17 q^{34} + q^{35} - 28 q^{36} + 36 q^{37} + 8 q^{38} - 37 q^{39} - 12 q^{40} - 3 q^{41} + 35 q^{42} - 10 q^{43} - 10 q^{44} + 5 q^{45} + 34 q^{46} + 18 q^{47} - 41 q^{48} - 45 q^{49} - 2 q^{50} - 51 q^{51} + 40 q^{52} - 16 q^{53} + 45 q^{54} - q^{55} + 20 q^{56} - 56 q^{57} - 4 q^{58} - 2 q^{59} + 3 q^{60} + 7 q^{61} - 92 q^{62} - 7 q^{63} + 4 q^{64} - 7 q^{65} + 42 q^{66} + 86 q^{67} + 17 q^{68} + 4 q^{69} + 2 q^{71} + 82 q^{72} + 4 q^{73} - 24 q^{74} + q^{75} + 16 q^{76} + 10 q^{77} - 59 q^{78} + 38 q^{79} - 36 q^{80} - 6 q^{81} - 11 q^{82} + 3 q^{83} + 133 q^{84} - 12 q^{85} + 14 q^{86} - 18 q^{87} - 64 q^{88} - 18 q^{89} - 13 q^{90} - 44 q^{91} - 138 q^{92} + 24 q^{93} - 42 q^{94} + 4 q^{95} + 52 q^{96} + 86 q^{97} + 5 q^{98} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
391.1 −2.63496 −0.435754 + 1.67634i 4.94302 −0.500000 − 0.866025i 1.14819 − 4.41709i −2.04731 − 3.54605i −7.75473 −2.62024 − 1.46094i 1.31748 + 2.28194i
391.2 −2.61548 −1.73157 + 0.0409042i 4.84073 −0.500000 − 0.866025i 4.52888 − 0.106984i 0.153914 + 0.266588i −7.42987 2.99665 − 0.141657i 1.30774 + 2.26507i
391.3 −2.43298 1.27908 + 1.16789i 3.91941 −0.500000 − 0.866025i −3.11198 − 2.84145i 0.991069 + 1.71658i −4.66990 0.272084 + 2.98764i 1.21649 + 2.10703i
391.4 −2.43179 −0.572849 − 1.63458i 3.91361 −0.500000 − 0.866025i 1.39305 + 3.97495i 2.25410 + 3.90422i −4.65350 −2.34369 + 1.87273i 1.21590 + 2.10599i
391.5 −2.16456 0.372160 − 1.69160i 2.68534 −0.500000 − 0.866025i −0.805564 + 3.66157i −2.16010 − 3.74141i −1.48347 −2.72299 − 1.25909i 1.08228 + 1.87457i
391.6 −2.13650 0.622113 − 1.61647i 2.56461 −0.500000 − 0.866025i −1.32914 + 3.45358i −0.884058 − 1.53123i −1.20630 −2.22595 − 2.01125i 1.06825 + 1.85026i
391.7 −2.10548 1.45422 − 0.940874i 2.43303 −0.500000 − 0.866025i −3.06183 + 1.98099i 1.04113 + 1.80328i −0.911744 1.22951 − 2.73647i 1.05274 + 1.82340i
391.8 −1.83243 −1.38289 − 1.04289i 1.35781 −0.500000 − 0.866025i 2.53405 + 1.91102i −1.30764 − 2.26489i 1.17678 0.824762 + 2.88440i 0.916216 + 1.58693i
391.9 −1.81264 −0.648519 + 1.60606i 1.28567 −0.500000 − 0.866025i 1.17553 − 2.91121i 2.07261 + 3.58987i 1.29483 −2.15885 − 2.08312i 0.906321 + 1.56979i
391.10 −1.72906 0.776870 + 1.54805i 0.989660 −0.500000 − 0.866025i −1.34326 − 2.67668i −0.504662 − 0.874101i 1.74694 −1.79294 + 2.40528i 0.864532 + 1.49741i
391.11 −1.48355 1.72850 + 0.110902i 0.200935 −0.500000 − 0.866025i −2.56432 − 0.164529i 0.647075 + 1.12077i 2.66901 2.97540 + 0.383387i 0.741777 + 1.28480i
391.12 −1.31616 −1.66711 + 0.469818i −0.267722 −0.500000 − 0.866025i 2.19419 − 0.618356i −0.473565 − 0.820239i 2.98469 2.55854 − 1.56648i 0.658080 + 1.13983i
391.13 −1.13527 1.73056 − 0.0717593i −0.711156 −0.500000 − 0.866025i −1.96466 + 0.0814664i −2.09824 − 3.63425i 3.07790 2.98970 − 0.248368i 0.567636 + 0.983175i
391.14 −0.904479 0.677895 − 1.59388i −1.18192 −0.500000 − 0.866025i −0.613142 + 1.44163i 0.997166 + 1.72714i 2.87798 −2.08092 − 2.16097i 0.452240 + 0.783302i
391.15 −0.786645 −1.72119 − 0.193698i −1.38119 −0.500000 − 0.866025i 1.35396 + 0.152372i 2.00060 + 3.46514i 2.65980 2.92496 + 0.666782i 0.393323 + 0.681255i
391.16 −0.721903 −1.07812 − 1.35560i −1.47886 −0.500000 − 0.866025i 0.778300 + 0.978613i 0.122429 + 0.212054i 2.51140 −0.675307 + 2.92301i 0.360952 + 0.625187i
391.17 −0.534539 1.03603 + 1.38804i −1.71427 −0.500000 − 0.866025i −0.553798 − 0.741959i 1.99149 + 3.44936i 1.98542 −0.853283 + 2.87609i 0.267269 + 0.462924i
391.18 −0.401048 −0.303467 + 1.70526i −1.83916 −0.500000 − 0.866025i 0.121705 − 0.683890i −1.84706 − 3.19920i 1.53969 −2.81582 − 1.03498i 0.200524 + 0.347318i
391.19 −0.208451 1.73142 − 0.0468820i −1.95655 −0.500000 − 0.866025i −0.360915 + 0.00977258i 0.482573 + 0.835840i 0.824745 2.99560 − 0.162345i 0.104225 + 0.180523i
391.20 −0.00951849 −1.39381 + 1.02824i −1.99991 −0.500000 − 0.866025i 0.0132670 − 0.00978731i 1.15776 + 2.00530i 0.0380731 0.885439 − 2.86636i 0.00475925 + 0.00824326i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 391.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.l.a yes 80
9.c even 3 1 855.2.j.b ✓ 80
19.c even 3 1 855.2.j.b ✓ 80
171.h even 3 1 inner 855.2.l.a yes 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
855.2.j.b ✓ 80 9.c even 3 1
855.2.j.b ✓ 80 19.c even 3 1
855.2.l.a yes 80 1.a even 1 1 trivial
855.2.l.a yes 80 171.h even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 2 T_{2}^{39} - 58 T_{2}^{38} + 112 T_{2}^{37} + 1545 T_{2}^{36} - 2868 T_{2}^{35} + \cdots + 27 \) acting on \(S_{2}^{\mathrm{new}}(855, [\chi])\). Copy content Toggle raw display